# What is the probability of the random occurrence of a value between 56 and 61 from a normally distributed population with mean 62 and standard deviation 4.5?

Question
Random variables
What is the probability of the random occurrence of a value between 56 and 61 from a normally distributed population with mean 62 and standard deviation 4.5?

2021-02-01

Step 1
The normal distribution:
A continuous random variable X is said to follow normal distribution with mean $$\displaystyle\mu$$ and standard deviation $$\displaystyle\sigma$$ if the probability density function of X is,
$$\displaystyle{f{{\left({x}\right)}}}={\frac{{{1}}}{{\sigma\sqrt{{{2}\pi}}}}}{e}^{{{\frac{{{\left({x}-\mu\right)}^{{{2}}}}}{{{2}\sigma^{{{2}}}}}}}},-\propto{<}{x}{<}\propto,-\propto{<}\mu{<}\propto,\sigma^{{{2}}}{>}{0}$$
Step 2
Finding the probability of the random occurrence of a value between 56 and 61:
The random occurrence of a value (X) follows a normal distribution with mean 62 and standard deviation of 4.5.The required probability is calculated as follows:
$$\displaystyle{P}{\left({56}{<}{X}{<}{61}\right)}={P}{\left({X}{<}{61}\right)}-{P}{\left({X}{<}{56}\right)}$$
$$=0.4121-0.0912=0.3209 \begin{bmatrix}Using\ Excel\ formula, \\"NORM.DIST(61,62,4,5,1)"\\"NORM.DIST(56,62,4,5,1)" \end{bmatrix}$$

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